Identifying communities plays a central role in understanding the structure of large networks. As practitioners analyze progressively larger networks, it becomes increasingly important to understand the computational complexity of candidate algorithms. We examine the complexity of the link clustering algorithm for overlapping community detection. We provide new, tight bounds for the original implementation and propose modifications to reduce algorithmic complexity. These new bounds are a function of the number of wedges in the graph. Additionally, we demonstrate that for several community detection algorithms, wedges predict runtime better than commonly cited graph features. We conclude by proposing a method to reduce the wedges in a graph by removing high-degree vertices from the network, identifying communities with an optimized version of link clustering, and heuristically matching communities with the removed vertices as a post-processing step. We empirically demonstrate a large reduction in processing time on several common data sets.